Using decomposition groups to prove theorems about quadratic residues

2020 ◽  
Vol 1 (1) ◽  
pp. 12-20
Author(s):  
Tomas Perutka
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Class Field Theory ◽  
Legendre Symbol ◽  
Quadratic Reciprocity ◽  
Class Field ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Quadratic Residues ◽  
Cyclotomic Extensions

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

A Generalization of Global Class Field Theory

1969 ◽  
Vol 21 ◽  
pp. 609-614
Author(s):  
Tae Kun Seo ◽  
G. Whaples
Keyword(s):  
Field Theory ◽  
Finite Field ◽  
Class Group ◽  
Rational Functions ◽  
Class Field Theory ◽  
Usual Sense ◽  
Finite Degree ◽  
Class Field ◽  
Reciprocity Law

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

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